Solve 6x^4-5x^3+7x^2-5x+1

$$6 x ^ { 4 } - 5 x ^ { 3 } + 7 x ^ { 2 } - 5 x + 1$$

Factor

$$\left(2x-1\right)\left(3x-1\right)\left(x^{2}+1\right)$$

Solution Steps

To factor the expression, solve the equation where it equals to 0.

$$6x^{4}-5x^{3}+7x^{2}-5x+1=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 1 and q divides the leading coefficient 6. List all candidates \frac{p}{q}.

$$±\frac{1}{6},±\frac{1}{3},±\frac{1}{2},±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=\frac{1}{3}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 6x^{4}-5x^{3}+7x^{2}-5x+1 by 3\left(x-\frac{1}{3}\right)=3x-1 to get 2x^{3}-x^{2}+2x-1. To factor the result, solve the equation where it equals to 0.

$$2x^{3}-x^{2}+2x-1=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -1 and q divides the leading coefficient 2. List all candidates \frac{p}{q}.

$$±\frac{1}{2},±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=\frac{1}{2}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 2x^{3}-x^{2}+2x-1 by 2\left(x-\frac{1}{2}\right)=2x-1 to get x^{2}+1. To factor the result, solve the equation where it equals to 0.

$$x^{2}+1=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 0 for b, and 1 for c in the quadratic formula.

$$x=\frac{0±\sqrt{0^{2}-4\times 1\times 1}}{2}$$

Do the calculations.

$$x=\frac{0±\sqrt{-4}}{2}$$

Polynomial x^{2}+1 is not factored since it does not have any rational roots.

$$x^{2}+1$$

Rewrite the factored expression using the obtained roots.

$$\left(2x-1\right)\left(3x-1\right)\left(x^{2}+1\right)$$

Evaluate

$$\left(2x-1\right)\left(3x-1\right)\left(x^{2}+1\right)$$